2016/08/17 by Stefan Steinerberger, Steinerberger, Stefan
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Inequalities and Applications #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.1608.05050
openalex publication_date 2016/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A version of the Cauchy-Schwarz inequality in operator theory is the following: for any two symmetric, positive definite matrices A,B ∈ ℝn × n and arbitrary X ∈ ℝn × n ‖AXB‖ ≤ ‖A2 X‖(1)/(2) ‖X B2‖(1)/(2). This inequality is classical and equivalent to the celebrated Heinz-Löwner, Heinz-Kato and Cordes inequalities. We characterize cases of equality: in particular, after factoring out the symmetry coming from multiplication with scalars ‖A2 X‖ = 1 = ‖X B2‖, the case of equality requires that A and B have a common eigenvalue λi = μj. We also derive improved estimates and show that if either λi λj = μk2 or λi2 = μj μk does not have a solution, i.e. if d > 0 where d amp;= min1 ≤ i,j,k ≤ n \ | log λi + log λj - 2log μk|:λi, λj ∈ σ(A), μk ∈ σ(B) \ amp;+min1 ≤ i,j,k ≤ n\ | 2logλi - log μj - logμk |:λi ∈ σ(A), μj, μk ∈ σ(B) \, then there is an improved inequality ‖AXB‖ ≤ (1 - cn,d)‖A2 X‖(1)/(2) ‖X B2‖(1)/(2) for some cn,d > 0 that only depends only on n and d. We obtain similar results for the McIntosh inequality and the Cordes inequality and expect the method to have many further applications.